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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Geometry of graph varieties

Author: Jeremy L. Martin
Journal: Trans. Amer. Math. Soc. 355 (2003), 4151-4169
MSC (2000): Primary 05C10, 14N20; Secondary 05B35
Published electronically: May 15, 2003
MathSciNet review: 1990580
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Abstract: A picture $\mathbf{P}$ of a graph $G=(V,E)$ consists of a point $\mathbf{P}(v)$ for each vertex $v \in V$ and a line $\mathbf{P}(e)$ for each edge $e \in E$, all lying in the projective plane over a field $\mathbf k$ and subject to containment conditions corresponding to incidence in $G$. A graph variety is an algebraic set whose points parametrize pictures of $G$. We consider three kinds of graph varieties: the picture space $\mathcal{X}(G)$ of all pictures; the picture variety $\mathcal{V}(G)$, an irreducible component of $\mathcal{X}(G)$ of dimension $2\vert V\vert$, defined as the closure of the set of pictures on which all the $\mathbf{P}(v)$ are distinct; and the slope variety $\mathcal{S}(G)$, obtained by forgetting all data except the slopes of the lines $\mathbf{P}(e)$. We use combinatorial techniques (in particular, the theory of combinatorial rigidity) to obtain the following geometric and algebraic information on these varieties:

a description and combinatorial interpretation of equations defining each variety set-theoretically;
a description of the irreducible components of $\mathcal{X}(G)$;
a proof that $\mathcal{V}(G)$ and $\mathcal{S}(G)$ are Cohen-Macaulay when $G$ satisfies a sparsity condition, rigidity independence.
In addition, our techniques yield a new proof of the equality of two matroids studied in rigidity theory.

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Additional Information

Jeremy L. Martin
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112
Address at time of publication: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Keywords: Graphs, graph varieties, configuration varieties
Received by editor(s): June 27, 2002
Received by editor(s) in revised form: January 28, 2003
Published electronically: May 15, 2003
Article copyright: © Copyright 2003 American Mathematical Society